Degenerations of varieties with torus actions

Toric varieties build an important class in algebraic geometry as their high symmetry allows to translate algebraic geometric properties into combinatorics and vice versa. Constructing toric degenerations of a variety enables us to also give combinatorial descriptions of geometric invariants of (non-toric) varieties.

In this talk we discuss two approaches to obtain toric degenerations: In the first part of the talk we use the Khovanskii bases based approach to obtain toric degenerations of varieties with torus action of complexity one. Afterwards we discuss how to obtain toric degenerations based on the theory of Cox rings.